Issue 59

N. Amoura et al, Frattura ed Integrità Strutturale, 59 (2022) 243-255; DOI: 10.3221/IGF-ESIS.59.18

1 ( )

 

 



 k P S P Q u Q d Q D P Q t Q d Q   ( , ) ( ) ( )  ( , ) ( ) ( ) ijk k ijk

(8)

2 ij

Where  stands for Hadamard Finite Part integral. Multiplying both sides of Eqn. 8 by the normal components at the source point   j n P leads to the traction equation

1 ( )

 

 

 

 

 k t P n P S P Q u Q d Q n P D P Q t Q d Q   ( , ) ( ) ( )  ( , ) ( ) ( ) j ijk k j ijk

(9)

2 i

The dual boundary element method eliminates need of domain sub-division and reduces both the computational effort and the perturbations due to mesh adjustments, since only domain boundaries are meshed. The finite elements used for meshing are of two types: regular elements and singular ones. The regular elements are isoparametric triangular elements with six nodes employed to mesh the domain boundaries. The singular elements, used to mesh the two crack surfaces, have their nodes shifted towards the center of the element in order to satisfy, smoothness at the boundary nodes, continuity of the displacement derivatives and boundary curvatures at these points; these conditions are required for the evaluation of the improper integrals inherent in a dual boundary equation formulation [24]. The dual equations Eqn. 7 and 9 are applied to the elements of each side of the crack to ensure a non-singular system to be solved. After evaluation of integrals over each element, a discretized system of equations is obtained and solved to get the unknown tensions and displacements. Afterwards, the strains at the selected sensor points are post-processed. The numerical process of 3-D DBEM have been implemented in the KSP software, developed by H. Kebir from the University of Technology of Compiegne. The software uses C++ programming language to solve the direct problem consisting of computing stresses and strains on the domain frontiers for a cracked structure subjected to a mixed field of boundary conditions (displacements and tractions). In this present study, we have implemented both the Low-Discrepancy Sequence (LDS) algorithm and the Nelder-Mead Simplex Algorithm (NMSA) to solve the inverse problem of crack identification. The inverse problem uses the computed (or measured) strains at some sensor points from a real case of cracked structure and minimizes the regularized objective function Eqn. 3. ased on the works of Box [25] and Spendley [26], the Nelder-Mead [17] version of the Simplex Algorithm (NMSA) is an optimization method applied to construct a non-local linear approximation of a n-dimensional constrained problem from a set of points at sufficient intervals. The method finds a locally optimal solution to an objective function (OF) in  n when it varies smoothly. At each iteration, the NMSA produces a new test position using four operations named: reflection, expansion, contraction and the similarity transformation. Through these operations, the algorithm extrapolates the behavior of the OF measured at each test point and arranged as a simplex. The algorithm then rejects the position which gives the highest value of the OF with the new test point and so the algorithm progresses [27]. The first step is to change the worst point with a point reflected through the centroid of the opposite side of the Simplex. If this point is better than the best current one, then we can attempt extending exponentially out along this line. Now, if this new point do not give a much better value for the OF than the previous value, then we contract the simplex towards the best point (Fig. 3). Additional constraints must be considered on domain limits to ensure embedded cracks, on the first guess for the crack’s identity to start the minimization process, which must converge to the actual solution as well as a convergence test to stop the progression of the algorithm. The first 3D example treated is relating to a cylindrical shaft in axial traction with an embedded elliptical crack. The dimensions used for the shaft are L/D=2 ; L and D , are the length and the diameter of the shaft respectively ; a/b=2 and D/a=10 , a and b the crack radii. The material has an elasticity modulus E= 200 GPa and a Poisson’s ratio ν =0.3. The NMSA was applied starting with an initial crack position close to the actual one to carry out a fast convergence and avoiding local minima. Convergence to the actual position of the crack (Fig. 4) was achieved in less than 1100 iterations with a normalized OF less than 10 -5 . Figs. 5-7 show the convergence of the normalized OF and the crack’s identity parameters to B N ELDER -M EAD OPTIMIZATION ALGORITHM

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